\subsection*{2012-07-16}

\begin{itemize}
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\item fluid mechanics is $\partial_\mu T^{\mu \nu}=0$ with the appropriate choice of $T^{\mu \nu}$. For a perfect
fluid $T^{\mu \nu}=(\rho+p) u^{\mu} u^{\nu}+p g^{\mu \nu}$. The conservation equation is not enough for describing
the dynamics of the system. We also need to have an equation of state which is of the form $f(p,\rho,\overrightarrow{u})=0$.
 If we solve the conservation equation for a prefect
fluid up to the first order in time derivatives with $\rho=const.$(non-compressibility) we get the Navier-Stokes equations. 
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\item Assuming that $\overrightarrow{u}$ is small, the conservation equation of a perfect fluid in falt spacetime gives:
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\begin{equation}
g^{\mu \nu}\partial_{\mu}p +\partial_{t} [(p+\rho) u^{\nu}]+(p+\rho)u^{\nu}(\partial_{i}u^i)+u^{\nu}
\partial_{i} [(p+\rho)u^{\nu}]=0
\end{equation}
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\item The conservation equation along with the equation of state can be written in a more direct form of equations like 
$\partial_t(variable)=\partial_{i}^2(variable)$. Solving these equations numerically is very efficient and we can plug the
answer into the GR side to see the evolution of the black hole(brane) horizon. This is in fact the main part of our job!

\item Consider the differential equation $\phi^{''}+\phi^2=7$. We need two boundary conditions to solve this.
If we have $\phi(0)$ and $\phi(0)^{'}$ then we only need to start from $x=0$ and get $\phi$ for all $x$ in
an iterative way. But if we have $\phi(0)$ and $\phi(1)$ we can't do that. One method which is so-called the shooting
method would be to shoot solutions from $x=0$ with different angles until we hit $\phi(1)$ at $x=1$. This takes
a long time and is not tractable for higher order differential equations. Instead we can solve this by assigning $\phi_{i}$ to
the point $x=\frac{i}{N}$ and then creating the first and higher order derivatives as algebraic expressions
(e.g. $\phi^{''}_{i}=N^2(\phi_{i+1}+\phi_{i-1}-2\phi_{i})$). Then we solve the whole thing as a system of equations.
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\item What is turbulence? There has been attemps to define it mathematically. Here's three of them:
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\begin{itemize}
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\item Consider releasing probes in a fluid and finding their pathways. If they diverged in a chaotic way there is
turbulence. This method is not ideal because it depends on the choice of the probes' locations and also it's
hard to quantify turbulence this way.
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\item Another method which is a variant of the method above is to consider an ensemble of initial conditions, evolve them
in time and do the same as above to define turbulence. This method has the same imperfections as the above method.
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\item  The third method to define turbulence is through dissipation. Dissipation comes from non-linear
terms in the dynamical equations of fluids. A good characteristic for turbulence might be that a wave generated from a
source in the fluid will divide into patches of higher frequency and it dissipates on the way(dissipation gets larger as the
frequency grows) creating an energy cascade. I'm not sure why this is a bad definition for dissipation.
\end{itemize} 
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\subsection*{Next time}
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\begin{enumerate}
\item Derive the fluid dynamics equations of the form $\partial_t(variable)=\partial_{i}^2(variable)$.
\item solve  $\phi^{''}+\phi^2=7$ numerically with $\{\phi(0)=1, \phi(1)=1\}$
,$\{\phi(0)=1, \phi^{'}(1)=1\}$ and $\{\phi(0)=1, \phi(1)+\phi^{'}(1)=1\}$.
\item Read "Easy Turbulence" by Krzysztof Gawedzki.
\item The ultimate goal is to search for a solid definition for turbulence in the GR picture.
\end{enumerate}
\end{itemize}
 